We report muon spin rotation (${mu}$SR) experiments together with first-principles calculations on microscopic properties of superconductivity in the kagome superconductor LaRu$_3$Si$_2$ with $T_{rm c}$ ${simeq}$ 7K. We find that the calculated normal state band structure features a kagome flat band and Dirac as well as van Hove points formed by the Ru-$dz^2$ orbitals near the Fermi level. Below $T_{rm c}$, ${mu}$SR reveals isotropic type-II superconductivity, which is robust against hydrostatic pressure up to 2 GPa. Intriguingly, the ratio 2$Delta/k_{rm B}T_{rm c}$ ${simeq}$ 4.3 (where ${Delta}$ is the superconducting energy gap) is in the strong coupling limit, and $T_{rm c}$/$lambda_{eff}^{-2}$ (where ${lambda}$ is the penetration depth) is comparable to that of high-temperature unconventional superconductors. We also find that electron-phonon coupling alone can only reproduce small fraction of $T_{rm c}$ from calculations, which suggests other factors in enhancing $T_{rm c}$ such as the correlation effect from the kagome flat band, the van Hove point on the kagome lattice, and high density of states from narrow kagome bands. Our experiments and calculations taken together point to strong coupling and the unconventional nature of kagome superconductivity in LaRu$_3$Si$_2$.
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